In basic number theory, how many positive factors (including 1 and the number itself) does the integer 9321 have?

Introduction:
In this aptitude question on numbers and factors, you are asked to determine how many positive divisors a given integer has. This is a standard topic in basic number theory and is very useful for many competitive exams.

Given Data / Assumptions:

• The number under consideration is 9321.
• Only positive factors are considered.
• Both 1 and the number itself are counted as factors.

Concept / Approach:
To find the number of factors of a positive integer, we first express the number as a product of prime powers. If a number N has the prime factorization:
N = p1^a1 * p2^a2 * ... * pk^ak then the total number of positive divisors is:
(a1 + 1) * (a2 + 1) * ... * (ak + 1) So the main task is to factorize 9321 into primes and then apply this formula.

Step-by-Step Solution:
Step 1: Test divisibility by 3. 9 + 3 + 2 + 1 = 15, which is divisible by 3, so 9321 is divisible by 3. 9321 / 3 = 3107. Step 2: Check if 3107 is prime. We test divisibility by primes up to its square root. 3107 is not divisible by 2, 3, 5, 7, 11, etc., and we eventually find that it has no smaller prime factors, so 3107 is prime.
Thus, 9321 = 3^1 * 3107^1. Step 3: Apply the divisor count formula. For 3^1 * 3107^1, the exponents are 1 and 1. So: Number of factors = (1 + 1) * (1 + 1) = 2 * 2 = 4. But this is not correct because we missed another factorization step. Let us correct that.
Step 4: Refine factorization carefully. In fact, when factorized correctly, 9321 = 3 * 13 * 239.
So the exponents are all 1: 9321 = 3^1 * 13^1 * 239^1. Number of factors = (1 + 1) * (1 + 1) * (1 + 1) = 2 * 2 * 2 = 8.

Verification / Alternative Check:
We can list the divisors explicitly: 1, 3, 13, 39, 239, 717, 3107, 9321. Counting these gives exactly 8 factors, which matches our formula based result.

Why Other Options Are Wrong:
5, 4, 7, and 6 are all smaller than 8 and do not match the divisor count given by the prime factorization method. None of them correctly represent the total number of positive factors of 9321.

Common Pitfalls:
Students often stop factorization too early or forget that each prime factor exponent contributes plus one to the divisor count. Another common error is to miss a prime factor, which leads to an incorrect number of total factors.

Final Answer:
The integer 9321 has exactly 8 positive factors.